Sequence And Series Question 447
Question: If $ {\log_{e}}5,,{\log_{e}}(5^{x}-1) $ and $ {\log_{e}}( 5^{x}-\frac{11}{5} ) $ are in A.P then the values of x are
Options:
A) $ {\log_5}4and{\log_5}3 $
B) $ {\log_3}4and{\log_4}3 $
C) $ {\log_3}4and{\log_3}5 $
D) $ {\log_5}6and{\log_5}7 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ {\log_{e}}5+{\log_{e}}( 5^{x}-\frac{11}{5} )=2{\log_{e}}(5^{x}-1) $
$ \Rightarrow {5^{x+1}}-11=5^{2x}+1-2\times 5^{x}\Rightarrow 5^{2x}-{{7.5}^{x}}+12 $ $ =0 $ Let $ 5^{x}=t, $ $ t^{2}-7t+12=0\Rightarrow ,t=4,3 $ $ 5^{x}=4, $ $ 5^{x}=3 $ $ . \begin{matrix} {\log_5}5x={\log_5}4 \\ x={\log_5}4 \\ \end{matrix} \begin{vmatrix} {\log_5}5^{x}={\log_5}3 \\ x={\log_5}3 \\ \end{matrix} $
BETA
